Consider a space object orbiting the Earth in a Low Earth Orbit (LEO). This space object is subject to arbitrary time-varying disturbances, with unspecified dynamics but known bounds. An example of such a disturbance is atmospheric drag, which is a function of the altitude of the orbit, the shape of the space object, atmospheric density, etc. These disturbances modify the orbit of the space object and eventually, in the absence of orbit corrections, the space object may fall to the Earth. Tracking such an object and predicting the impact point is critical to prevent loss of life and important assets on Earth. Often such an object is observed by unsynchronized sensors which measure only line-of-sight (LOS) angles, such as azimuth and elevation. Such sensors may be, for example, cameras, infrared imagers, telescopic devices, and the like. That is, at any given time, only LOS direction measurements from a single sensor are available. These sensors may be located at various points on the surface of the Earth or in space. The observations are incomplete in that they do not provide range information, thus making it impossible to locate the position with one observation. However, with multiple non-synchronous observations from a single out-of-plane sensor, or from multiple sensors, one can obtain estimates of the position and velocity states of motion of the space object in the presence of disturbances that alter the orbit of the space object.
FIG. 1A illustrates a system 8 including a three-dimensional space 10 through which a target (i.e., object of interest) T moves along a trajectory 12 in a manner such that it occupies a first location or position at a time t1, a second location at a time t2, and a third location at a time t3. A first line-of-sight (LOS)-only sensor S1 is located so that the target's position at time t1 lies at an unknown range along a LOS direction û1. Such a sensor may be characterized as being “two-dimensional,” as it senses only the LOS direction through angle measurements, and does not provide range measurements. Similarly, a second two-dimensional, LOS-only sensor S2 is located so that the target position at time t2 lies at an unknown range along a LOS direction û2, and a third two-dimensional, LOS-only sensor S3 is located so that the target position at time t3 lies at an unknown range along a line-of-sight direction û3. Each of the sensors S1, S2, and S3 measures the LOS direction in three-dimensional space 10 of FIG. 1A along LOS lines û1, û2, and û3, respectively, at times t1, t2, and t3, respectively. Each sensor S1, S2, and S3 is capable of being slewed or pointed under control of a cueing direction control signal to aid in tracking a target. Thus, in the example of FIG. 1A, the target T has unknown dynamics and is at unknown range from each sensor. The number of sensors is three, and they are at known positions or locations. There are three reporting times, and those reporting times are known. The reporting times may be simultaneous or they may be in time sequence t1, t2, t3. Each sensor S1, S2, and S3 reports as a signal z the angular location of the target along a line-of-sight (LOS). In the arrangement of FIG. 1A, the single measurement provided by any one of the sensors S1, S2, and S3 at times t1, t2, and t3, respectively, is insufficient to provide a complete observation of the state of the target T. The information is incomplete because having only LOS direction in three dimensions is insufficient to establish range, and therefore location. While other measurements may occur, such as at times t2 and t3, to estimate position (location) and velocity with finite variance often requires a significant number of successive LOS direction measurements. The problems are exacerbated by the fact that the trajectory 12 taken by the target may be driven by unknown arbitrary time-varying inputs or disturbances, with unspecified dynamics but known bounds.
Information filters, as described by Y. Bar-Shalom, X. R. Li, and T. Kirubarajan, Estimation with Applications to Tracking and Navigation: Theory, Algorithms, and Software; New York, N.Y.: Wiley, 2001 are used in the prior art to aid in resolving problems associated with insufficient measurement. Such information filters use information matrices as opposed to covariance matrices, and are an outgrowth of Kalman filters. Covariance matrices are the inverses of information matrices whenever the inverses exist. The information filter founded on Kalman filtering has disadvantages in the context of the situation of FIG. 1A. The first disadvantage arises from the fact that the trajectory of the target may not be a path subject to dynamic modeling, but may be arbitrary, and thus such an information filter may not track properly unless an optimal process noise covariance can be chosen to model the target motion. Second, the information filter in the prior art does not always provide a trustworthy covariance matrix that accounts for the actual estimation errors. This difficulty stems from a choice of process noise covariance that may not be optimal for modeling the target motion.
Prior art includes the use of Optimal Reduced State Estimation (ORSE) using sensors which generate three-dimensional measurements of targets driven by unknown input parameters within known bounds, as described in U.S. Pat. No. 7,180,443, issued Feb. 20, 2007 in the names of Mookerjee and Reifler, and entitled Reduced State Estimator for Systems with Physically Bounded Parameters. This patent describes how to determine state estimates and state error covariance for generalized or arbitrary motion of a target or moving object where the sensors provide complete measurements. Complete measurement in this context means measurements locating a point in three dimensional space at a known time. Since the measurements are nominally complete as to angle and range from the sensor, the target location is immediately established with a well-conditioned covariance matrix. ORSE processing provides many advantages over the Kalman filter for systems in which the input parameters are unknown except for the known bounds.
Improved or alternative processing is desired for estimating systems with arbitrarily time varying, but bounded, input parameters using sensor measurements which are incomplete or whose covariance matrices are ill-conditioned.